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Optimal order time discretizations for stochastic semilinear wave equations with multiplicative noise

Xiaobing Feng, Yukun Li, Liet Vo

TL;DR

The paper tackles numerical approximation of the stochastic semilinear wave equation with multiplicative noise on a bounded domain by introducing two implicit time-discretization schemes in mixed form. Through a blend of stochastic Taylor-type constructions and structure-based analysis, the authors prove energy stability for both schemes and derive sharp temporal convergence rates: a linear-in-time (in the energy norm) rate for the first scheme and an optimal $O(\tau^{3/2})$ rate in the $L^2$-norm for the displacement with the second scheme. The analyses hinge on enhanced Hölder continuity results and careful handling of the nonlinear noise, supported by numerical experiments that validate the sharpness of the theoretical error estimates. The work advances practical, provably optimal time discretizations for stochastic wave equations with nonlinear drift and diffusion terms, with implications for simulations in uncertain media.

Abstract

This paper is concerned with developing and analyzing two novel implicit temporal discretization methods for the stochastic semilinear wave equations with multiplicative noise. The proposed methods are natural extensions of well-known time-discrete schemes for deterministic wave equations, hence, they are easy to implement. It is proved that both methods are energy-stable. Moreover, the first method is shown to converge with the linear order in the energy norm, while the second method converges with the $\mathcal{O}(τ^{\frac32})$ order in the $L^2$-norm, which is optimal with respect to the time regularity of the solution to the underlying stochastic PDE. The convergence analyses of both methods, which are different and quite involved, require some novel numerical techniques to overcome difficulties caused by the nonlinear noise term and the interplay between nonlinear drift and diffusion. Numerical experiments are provided to validate the sharpness of the theoretical error estimate results.

Optimal order time discretizations for stochastic semilinear wave equations with multiplicative noise

TL;DR

The paper tackles numerical approximation of the stochastic semilinear wave equation with multiplicative noise on a bounded domain by introducing two implicit time-discretization schemes in mixed form. Through a blend of stochastic Taylor-type constructions and structure-based analysis, the authors prove energy stability for both schemes and derive sharp temporal convergence rates: a linear-in-time (in the energy norm) rate for the first scheme and an optimal rate in the -norm for the displacement with the second scheme. The analyses hinge on enhanced Hölder continuity results and careful handling of the nonlinear noise, supported by numerical experiments that validate the sharpness of the theoretical error estimates. The work advances practical, provably optimal time discretizations for stochastic wave equations with nonlinear drift and diffusion terms, with implications for simulations in uncertain media.

Abstract

This paper is concerned with developing and analyzing two novel implicit temporal discretization methods for the stochastic semilinear wave equations with multiplicative noise. The proposed methods are natural extensions of well-known time-discrete schemes for deterministic wave equations, hence, they are easy to implement. It is proved that both methods are energy-stable. Moreover, the first method is shown to converge with the linear order in the energy norm, while the second method converges with the order in the -norm, which is optimal with respect to the time regularity of the solution to the underlying stochastic PDE. The convergence analyses of both methods, which are different and quite involved, require some novel numerical techniques to overcome difficulties caused by the nonlinear noise term and the interplay between nonlinear drift and diffusion. Numerical experiments are provided to validate the sharpness of the theoretical error estimate results.
Paper Structure (15 sections, 8 theorems, 99 equations, 3 tables)

This paper contains 15 sections, 8 theorems, 99 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Useful facts
  5. Structure assumptions
  6. Variational solutions and their properties
  7. Time-discrete approximation schemes
  8. Formulation of schemes
  9. Stability estimates
  10. Error estimates
  11. Error estimates for the scheme with $\theta =0$
  12. Error estimates for the scheme with $\theta = \frac{1}{2}$
  13. Numerical Experiments
  14. Finite element spatial discretization
  15. Test examples

Key Result

Lemma 1

Let $\phi \in C^{1,\alpha} ([0, T]; \mathbb{R})$ for some $\alpha \in (0, 1]$. Then, there exists a constant $C_e >0$ such that where the constant $C_e$ satisfies

Theorems & Definitions (16)

  • Lemma 1
  • Lemma 2
  • Definition 3
  • Lemma 4
  • Remark 1
  • Lemma 1
  • Proof 1
  • Lemma 1
  • Proof 2
  • Lemma 2
  • ...and 6 more